Theoretical model of the optimum flock size of birds flying in formation

Abstract

Although previous models of formation flight in birds show decreased flight costs relative to solo flight, they do not explain flock size variation across species or at different times of the year for a given species. This model captures some of the diversity in flock size observed in nature by incorporating energetic costs of flight and energy income from foraging. It turns out that within a myriad of possible flock sizes there is one that is optimal for maximizing energetic efficiency (net energetic gain/energy expenditure) for a given maximum range speed under given conditions (i.e. flight distance and total time to complete the journey). Net energetic gain from foraging equals the rate of prey encountered times the time spent foraging. Energy expenditure in flight, as derived from an approximation technique, is represented in close-form. This expression is a function of flock size and wing-tip spacing (WTS) and simplifies flight cost calculations. For certain WTS, a good approximation to the induced drag for a member of a flock of size n is 1/ n th of the induced drag of a single bird. The optimum flight speed of a flock is (1/ n ) 1/4 of that of a single bird. Optimum flock size in migrating birds is predicted and results are compared to observations made under various conditions. If migration is constrained by daylight hours, seasonal variation in flock size is expected if the start time of the north and southward migration are equidistant with respect to the summer solstice (21 June). Under certain conditions, such as long non-stop migration, solo flight is an optimum migratory strategy.